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Concept
Inégalité de Bernoulli
Résumé
In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of . It is often employed in real analysis. It has several useful variants:
for every integer and real number . The inequality is strict if and .
for every integer and every real number .
for every even integer and every real number .
for every real number and . The inequality is strict if and .
for every real number and .
Jacob Bernoulli first published the inequality in his treatise "Positiones Arithmeticae de Seriebus Infinitis" (Basel, 1689), where he used the inequality often.
According to Joseph E. Hofmann, Über die Exercitatio Geometrica des M. A. Ricci (1963), p. 177, the inequality is actually due to Sluse in his Mesolabum (1668 edition), Chapter IV "De maximis & minimis".
The first case has a simple inductive proof:
Suppose the statement is true for :
Then it follows that
Bernoulli's inequality can be proved for case 2, in which is a non-negative integer and , using mathematical induction in the following form:
we prove the inequality for ,
from validity for some r we deduce validity for .
For ,
is equivalent to which is true.
Similarly, for we have
Now suppose the statement is true for :
Then it follows that
since as well as . By the modified induction we conclude the statement is true for every non-negative integer .
By noting that if , then is negative gives case 3.
The exponent can be generalized to an arbitrary real number as follows: if , then
for or , and
for .
This generalization can be proved by comparing derivatives. The strict versions of these inequalities require and .
Instead of the inequality holds also in the form where are real numbers, all greater than , all with the same sign. Bernoulli's inequality is a special case when . This generalized inequality can be proved by mathematical induction.
In the first step we take . In this case the inequality is obviously true.
In the second step we assume validity of the inequality for numbers and deduce validity for numbers.
Source officielle
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